subject

math.DG

math.DG: explore 8 source-linked works published from 2025 to 2026, with original documents and citations.

This collection is a preview while coverage and quality are evaluated.

Coverage and selection

Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-15. Counts describe this index, not the complete source archives.

arXiv2026

Transversality Conditions for Boundary Constraints Defined by Differential Equations

What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.

math.OC↗

arXiv2026

The Discrete Harmonic Center of a Quadrilateral

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

math.DG↗

arXiv2026

Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements

Determining the minimal number of intensity measurements required for phase retrieval in $\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].

quant-ph↗

arXiv2026

Accelerate Vector Diffusion Maps by Landmarks

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

stat.ML↗

arXiv2026

Cultural Bias Without a Cultural Self:A Disassociation Study of LLM's Persona and Bias

Language models prompted with cultural personas increasingly stand in for human respondents in cross-cultural research. Their responses separate personas cleanly, and that separation is read as evidence of a cultural point of view. We show that the separation is real, that the point of view is not, and that one criterion tells them apart. A trait is structure internal to one respondent that survives a change of measurement frame; a bias needs only group-specific item means. To test for the first, we represent a single response set as an Item--Dimension matrix and treat its correlation matrix as a point on the manifold of symmetric positive definite matrices. In humans this carries what a trait should: it reproduces across test--retest sessions sharing no items, order or context ($r=0.77$, $N=89$); on public NEO-PI-R data it identifies individuals at up to $76\%$ against a $0.4\%$ chance level ($N=263$); and it predicts GPA ($R^2=0.281$, $p=0.003$) where BigFive aggregates from the same responses predict nothing ($R^2=0.018$). In four frontier LLMs it returns nothing. Persona structure is readable only while every instance shares one item order: give each its own order and separation falls from $94.7\%$ to chance, while realigning instances to \emph{any} shared random order restores it to $82$--$84\%$. Responses generated independently item by item, with no latent structure, reproduce the entire pattern. The cultural signal is a group template, not a property of any instance, and alignment regimes differ only in which stereotype survives on the surface.

stat.ML↗

arXiv2026

Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses

Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.

math.NA↗

arXiv2025

Momentum-based gradient descent methods for Lie groups

Polyak's Heavy Ball (PHB; Polyak, 1964), a.k.a. Classical Momentum, and Nesterov's Accelerated Gradient (NAG; Nesterov, 1983) are well-established momentum-descent methods for optimization. Although the latter generally outperforms the former, primarily, generalizations of PHB-like methods to nonlinear spaces have not been sufficiently explored in the literature. In this paper, we propose a generalization of NAG-like methods for Lie group optimization. This generalization is based on the variational one-to-one correspondence between classical and accelerated momentum methods (Campos et al., 2023). We provide numerical experiments for chosen retractions on the group of rotations based on the Frobenius norm and the Rosenbrock function to demonstrate the effectiveness of our proposed methods, and that align with results of the Euclidean case, that is, a faster convergence rate for NAG.

math.OC↗

arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Transversality Conditions for Boundary Constraints Defined by Differential Equations	2026-09-03	2609.04084	arxiv
The Discrete Harmonic Center of a Quadrilateral	2026-09-01	2609.00917	arxiv
Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements	2026-08-31	2607.27719	arxiv
Accelerate Vector Diffusion Maps by Landmarks	2026-08-30	2603.21247	arxiv
Cultural Bias Without a Cultural Self:A Disassociation Study of LLM's Persona and Bias	2026-08-30	2607.02368	arxiv
Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses	2026-08-28	2608.28073	arxiv
Momentum-based gradient descent methods for Lie groups	2025-07-31	2404.09363	arxiv
Turing complete Navier-Stokes steady states via cosymplectic geometry	2025-07-10	2507.07696	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.